If asked how many factors a number has, what would you do? Could you count its factors without an explicit list? The answer may be found in a particular representation of numbers: prime factorization.
What is a prime factorization? Let's begin with factoring a number. To factor a number, like 6, show one way to get it from multiplication. For example we can get 6 from 1 × 6. We can get 6 from 2 × 3 as well. Thus, we say that 1 × 6 and 2 × 3 are factorizations of 6. We also say that 1, 2, 3, and 6 are factors of 6 because they appear in the factorizations of 6. To finalize the matter, we can write 6 = 1 × 6 and 6 = 2 × 3.
Are there any other factorizations of 6? You might say that 3 × 2 = 6 is another factorization of 6 and you'd be right. However, it does not lead to any new factors of 6. Are there any factors of 6 other than 1, 2, 3, and 6?
Notice that in the factorization 2 × 3 = 6, both factors are prime. We call such a factorization a "prime factorization". Is there any other way to factor 6 into primes?
For a given number n, how many factorizations does it have and how many of them are prime factorizations? Searching for the answer to that question will lead you to an understanding of a very important mathematical result: The Fundamental Theorem of Arithmetic. Let's try it!
Factorizations are a way of expressing numbers as configurations in n-dimensions. For example, there are two ways of arranging 12 into a rectangle and 3 ways of arranging it into a brick. Try this!